Download The Effects of Adjustment Costs and Uncertainty on Investment

The E¤ects of Adjustment Costs and Uncertainty on
Investment Dynamics and Capital Accumulation
Guiying Laura Wu
Nanyang Technological University
March 17, 2010
Abstract
This paper provides a uni…ed framework to study how capital adjustment costs
and uncertainty a¤ect investment dynamics and capital accumulation. It considers
an ongoing …rm with stochastic downward sloping demand curve and facing three
possible forms of adjustment costs: complete or partial irreversibility, …xed costs of
undertaking any investment and the traditional quadratic adjustment costs. The
quantities of interest are the impact e¤ects of demand shocks on capital adjustment
in the short run, and the expected capital stock level in the long run, under di¤erent
forms of adjustment costs, and at di¤erent levels of uncertainty.
JEL Classi…cation: E22, D92, C61
Key Words: Investment, Capital Adjustment Costs, Uncertainty
This paper is based on Chapter 3 of my DPhil thesis at University of Oxford. I would like to thank
my supervisors Steve Bond and Måns Söderbom for their helpful suggestions and invaluable support.
The usual disclaims apply. Address: Division of Economics, School of Humanities and Social Sciences,
Nanyang Technological University, Singapore, 637332. E-mail: [email protected].
1
1
Introduction
According to the static neoclassical producer theory, a …rm’s optimal investment is to
equalize the marginal revenue product of capital (MPK, hereafter) to the user cost of
capital (ucc, hereafter), as derived by Jorgenson (1963). Two important features of
investment turn this static problem into dynamic: uncertainty about future economic
environment and costly adjustment of capital stock. Without uncertainty, a …rm can
follow a deterministic optimal investment path even in the presence of adjustment costs.
Without adjustment costs, a …rm can instantaneously and costlessly its adjust capital
stock in each period even in the presence of uncertainty. Therefore studying the e¤ects of
uncertainty and adjustment costs on …rm’s investment decision and capital accumulation
is crucial in understanding investment.
This paper provides a uni…ed framework to study these short-run and long-run e¤ects
on an ongoing …rm. It considers a …rm facing stochastic downward sloping demand curve1
and three possible forms of adjustment costs— complete or partial irreversibility, …xed
costs of undertaking any investment, and the traditional quadratic adjustment costs2 .
By construction, in the absence of any adjustment costs, both investment dynamics and
capital accumulation are invariant to the level of uncertainty in such a framework. This
provides a useful benchmark to investigate two sequential questions: …rst, what are the
e¤ects of di¤erent forms of adjustment costs, compared with the frictionless benchmark;
and second, what are the e¤ects of uncertainty through these adjustment costs?
In the past three decades the investment literature has paid much attention to irreversibility and achieved important insights. When investment is irreversible, the optimal
investment policy is to purchase capital only as needed to prevent the MPK from rising
above an optimally derived hurdle. The hurdle, which is the ucc appropriately de…ned
to take account of irreversibility and uncertainty, is higher than the Jorgensonian ucc
and hence predicts less investment. This result is known as the ‘real option’e¤ect in the
option approach (Bertola,1988; Pindyck,1988; Dixit and Pindyck,1994); or the ‘user cost’
e¤ect in the q-approach (Abel and Eberly,1996); and is uni…ed in Abel, Dixit, Eberly and
Pindyck (1996). A related result is that an increase in uncertainty facing the …rm tends
to increase the ucc under irreversibility, which further reduces the optimal investment.
This relationship is formalized as a negative e¤ect of uncertainty on the impact e¤ect of
positive demand shocks on investment dynamics in Bond, Bloom and Van Reenen (2007)
and Bloom (2009).
Nevertheless, the negative short-run e¤ects do not necessarily imply lower capital
1
As emphasized in Caballero (1991), Pindyck (1993) and Abel and Eberly (1996), in analyzing the
e¤ect of uncertainty, it is important for the MPK of the …rm to be a decreasing function of the capital stock. Otherwise the future MPK are una¤ected by today’s investment, so the link from today’s
investment to future returns is broken.
2
In the early investment literature, such as Hayashi (1982), the adjustment costs merely referred
to what is called convex or quadratic adjustment costs more recently. Irreversibility stood as another
type of friction that a¤ects …rm’s investment decision. Fixed adjustment costs became the focus of
new interest since 1990s. Abel and Eberly (1994) include both traditional quadratic adjustment costs
and irreversibility, together with a …xed component of capital adjustment costs into an augumented
adjustment cost function. To distinguish from quadratic adjustment costs, irreversiblity and …xed costs
are sometimes called non-convex adjustment costs.
2
stock in the long run. This is because if the MPK is unusually low at date t, the …rm
would like to sell some of its capital at a positive price. However, under irreversibility, the
…rm cannot sell capital, and it is constrained by its own past investment behavior to have
a capital stock that is higher than it would choose if it could start fresh at date t. Abel
and Eberly (1999) refer to this e¤ect as the ‘hangover’e¤ect to indicate the dependence
of the current capital stock on past behavior.
In the special case of complete irreversibility, no depreciation, a Brownian motion
demand process and an in…nite time horizon, Abel and Eberly (1999) demonstrate analytically that the user cost e¤ect and hangover e¤ect have opposite implications for the
expected long-run capital stock. Irreversibility may increase or decrease capital accumulation relative to the frictionless benchmark. Furthermore, an increase in uncertainty can
either increase or decrease the long-run capital stock under irreversibility relative to that
under reversibility.
Recently more and more empirical evidence has highlighted the importance of other
forms of adjustment costs. For example, Cooper and Haltiwanger (2006) …nd the evidence
of …xed adjustment costs in plant level data; Eberly, Rebelo and Vincet (2008) emphasize
the importance of quadratic adjustment costs in …rm level data; and Bond, Söderbom and
Wu (2008) …nd the signi…cance of both …xed and quadratic adjustment costs at the …rm
level. However, compared with irreversibility, there is little theoretical work about how
uncertainty a¤ects …rm’s investment dynamics and capital accumulation in the presence
of …xed and quadratic adjustment costs. No analytical solution to the investment model
with generalized adjustment costs is the main reason for this gap. Lack of well-de…ned
distinct short-run and long-run quantities of interest is another cause of the gap.
Using numerical dynamic programming methods, this paper solves a generalized investment model with complete and partial irreversibility, …xed costs of investment and
quadratic adjustment costs. Following Bloom (2009), the impact e¤ect of positive demand shocks on capital adjustment is de…ned as the quantity of interest for the short-run
dynamics. Following Abel and Eberly (1999), the ratio of the expected capital stock level
with adjustment costs to the expected capital stock level without adjustment costs is
de…ned as the quantity of interest for the long-run accumulation.
Concerning the short-run e¤ects, the presence of complete irreversibility, partial irreversibility and quadratic adjustment costs all dampens the responsiveness of investment
to new information about demand. Furthermore, in the presence of complete irreversibility, partial irreversibility and …xed adjustment costs, the impact e¤ect of positive demand
shocks on capital adjustment is a non-increasing function of uncertainty. This con…rms
the …ndings in Bloom (2009) for partial irreversibility but also highlights the importance
of other forms of non-convex adjustment costs.
Concerning the long-run e¤ects, the numerical solution in this paper replicates the
analytical …nding in Abel and Eberly (1999) for the special case of complete irreversibility.
Similar properties are found for partial irreversibility, in the sense that the presence of
partial irreversibility could either increase or decrease the expected long-run capital stock
relative to that under frictionless case; and uncertainty does not ease the ambiguity but
rather deepens it. In contrast, in the presence of quadratic adjustment costs, the expected
long-run capital stock is unambiguously lower than the frictionless level, due to a ucc that
3
is higher than the Jorgensonian ucc. Furthermore, this user cost e¤ect gets stronger with
an increase in uncertainty hence further reduces the expected long-run capital stock at a
higher level of uncertainty. The …xed adjustment costs have the same e¤ect as quadratic
adjustment costs at complete certainty but the same e¤ect as partial irreversibility in an
uncertain environment.
The numerical methods also allow comparative statics about the e¤ects of other model
parameters on these …ndings. In particular, this paper examines the role of the demand
growth rate, the discount rate, the capital share in the production function, the demand
elasticity, the depreciation rate and the serial correlation parameter in a trend stationary
demand process.
The rest of the paper is organized as follows. Section 2 constructs an investment model
under uncertainty and characterizes the optimal investment decision in the presence of
di¤erent forms of adjustment costs. Section 3 investigates the e¤ects of adjustment costs
and uncertainty on both short-run capital adjustment and long-run capital accumulation. The e¤ects of other model parameters are presented in Section 4. Section 5 o¤ers
concluding remarks.
2
An Investment Model under Uncertainty
This section sets up an investment model for a …rm operating under uncertainty. The
functional forms are chosen strictly following Abel and Eberly (1999), except for two
variations. First, this section assumes a discrete rather than continuous timing in order
to solve the model using standard numerical methods. Second, it allows depreciation of
capital stock and three forms of capital adjustment costs. This model therefore nests Abel
and Eberly (1999) as a special benchmark case but is also general enough to incorporate
other cases in this literature.
2.1
Short-run Pro…t Optimization
Time is discrete and horizon is in…nite. By paying capital adjustment costs, new investb t immediately in period t, which depreciates
ment It contributes to productive capital K
at the end of each period.3
Assumption 1 Timing:
The law of motion for capital stock is
Kt+1 = (1
where
) (Kt + It )
is the constant depreciation rate.
3
(1
bt
)K
(1)
Compared with alternative lagged timing assumption, such as Kt+1 = (1
)Kt + It , and only
Kt is productive in period t, Assumption 1 does not a¤ect the qualitative implications of the model,
but allows for a closed-form solution to the investment problem in the frictionless case, which does not
involve any expectation term. This provides a convenient benchmark for studying the e¤ets of captial
adjustment costs. In the special case of continuous timing and no depreciation as assumed in Abel and
Eberly (1999), this timing di¤erence vanishes.
4
b t and labor Lt to produce nonstorable output
Consider a …rm that uses capital stock K
Qt , according to a nonstochastic constant returns to scale Cobb-Douglas technology.
Assumption 2 Production:
The production function is
Qt = Lt1
where the capital share
satis…es 0 <
< 1.
bt
K
(2)
The …rm faces an isoelastic, downward-sloping, stochastic demand curve. Denote Xt
as the random component in demand, which can be interpreted as changes in the quantity
demanded Qt for any given price of output Pt , and is called ‘horizontal demand shocks’
4
in Abel and Eberly (1999).
Assumption 3 Demand:
The demand schedule is
Qt = Xt Pt
where
"<
"
(3)
1 is the demand elasticity with respect to price.
The demand shift parameter Xt is the only source of uncertainty in this model. Abel
and Eberly (1999) assume Xt evolves exogenously according to a geometric Brownian
motion with mean t and variance 2 t. By Ito’s Lemma, this implies the log of Xt
follows a Brownian motion with mean (
0:5 2 ) t and variance 2 t. The discrete time
analogue of this process is described in the following assumption:
Assumption 4 Demand Stochastic:
The law of motion for Xt is
xt
where e =
0:5
2
> 0, et =
ln Xt = xt
t,
i:i:d:
t
1
+ e + et
(4)
N (0; 1) , and x0 = 0.
Firm making decisions in period t knows Xt and the parameter values of X0 , and
, but are uncertain about future levels of demand which depend on future realizations
of the demand shocks et . The standard deviation of these demand shocks therefore
measures the level of uncertainty faced by the …rm. The condition > 0:5 2 guarantees
that the MPK has a non-degenerate ergodic distribution, as restricted in the Eq. (7) of
Abel and Eberly (1999).
Labor is a variable input hence is adjusted instantaneously and costlessly. In each
period, for given capital stock and demand realization, the …rm chooses labor Lt to
maximize its instantaneous operating pro…t Pt Qt wLt , where w is a constant wage rate.
4
In the investment literature, Hartman (1972), Abel (1983) and Caballero (1991) highlight the e¤ect
of uncertainty on the expected investment expenditure through the curvature of MPK to the stochastic
variable that characterizes uncertainty. Compared with the alternative ‘vertical demand shocks’, within
this class of model, the speci…cation for the horizontal demand shocks e¤ectively isolate this HartmanAbel-Caballero e¤ect and allow this paper to focus on the e¤ect of uncertainty through the sole channel
of capital adjustment costs. Bond, Söderbom and Wu (2008) o¤er a structural estimation for the e¤ects
of uncertainty through both the Hartman-Abel-Caballero e¤ect and capital adjustment costs e¤ect.
5
Lemma 1 Operating Pro…t:
The maximized value of operating pro…t is given by
where
bt) =
(Xt ; K
0<
1
<
"
=
h
1
b t1
Xt K
1
1 + ("
and
h = (1
)
" 1
w
1)
(5)
(6)
<1
" 1
( ")
"
(7)
Proof: See the Eq. (3) in Abel and Eberly (1999).
2.2
Adjustment Cost Function
Besides production technology and demand conditions, …rm’s investment behavior also
depends on capital adjustment costs. This section models three forms of adjustment costs
that have been highlighted in the investment literature. Abel and Eberly (1994) provide
an extensive discussion about the economic rationale of these adjustment costs and the
appropriateness of the speci…cation.
2.2.1
Complete and Partial Irreversibility
The early irreversibility literature completely rules out the regime of negative gross investment, hence investment exhibits irreversibility. More recent research allows a wedge
between the purchase price of capital pI and the sale price of capital pS , as a result of
capital speci…city, or more generally, the adverse selection in the second-hand capital
goods market. Normalize the purchase price pI to one and denote bi = 1 pS > 0, so
that the parameter bi can be interpreted as the di¤erence between the purchase price and
the sale price expressed as a percentage of the purchase price. For example, pS = 0:8
gives bi = 0:2, indicating that the sale price is 20% lower than the purchase price.
Assumption 5 Irreversibility:
The functional form of irreversibility is
G(It ) =
bi It 1[It <0]
where 1[It <0] is an indicator equal to one if investment is strictly negative.
Letting pS = 0 or bi = 1 ensures the …rm will never disinvest. This corresponds to
the case of complete irreversibility. In contrast, partial irreversibility refers to the more
general case where 0 < bi < 1.
6
2.2.2
Fixed Costs
The …xed costs re‡ect indivisibilities in capital or increasing returns to scale of investment.
They are paid at each point of time if any non-zero investment is undertaken. One way
to model these costs is to assume them to be proportional to the operating pro…t.5
Under this speci…cation, …rst, these costs can be rationalized as pro…t loss due to the
interruption in production during periods of large adjustment; second, these costs do not
become irrelevant as the …rm grows larger.
Assumption 6 Fixed Costs:
The functional form of …xed costs is
G(Xt ; Kt ; It ) = bf 1[It 6=0]
t
where 1[It 6=0] is an indicator equal to one if investment is non-zero. t is de…ned in
equation (5). The parameter bf is interpreted as the fraction of operating pro…t loss due
to any non-zero investment.
2.2.3
Quadratic Adjustment Costs
Quadratic adjustment costs re‡ect those costs that increase convexly in the level of investment. The speci…cation considered here includes three features. First, the costs are
quadratic in investment rate, to re‡ect increasing marginal adjustment cost. Second, the
costs attain their minimum value of zero at zero investment, so that the …rm can avoid
these costs by setting investment equal to zero. Third, the level of these costs is proportional to capital stock, so that a given investment rate imposes costs that increase with
the size of the …rm, and do not become irrelevant as the …rm grows larger.
Assumption 7 Quadratic Adjustment Costs:
The functional form of quadratic adjustment costs is
G(Kt ; It ) =
bq
2
It
Kt
2
Kt
where bq measures the magnitude of quadratic adjustment costs.
The model allows for these three forms of adjustment costs, specifying the adjustment
cost function to be
G(Xt ; Kt ; It ) =
bi It 1[It <0] + bf 1[It 6=0]
5
t+
bq
2
It
Kt
2
Kt
(8)
This speci…cation follows Caballero and Engel (1999) and Bloom (2009). An alternative is to model
these …xed costs proportional to the capital stock, such as Caballero and Leahy (1996) and Abel and
Eberly (2001), so that G(Xt ; Kt ; It ) = bF 1[It 6=0] Kt , where bF is the fraction of capital stock loss due to
any non-zero investment. Cooper, Haltiwanger and Power (1999) and Cooper and Haltiwanger (2006)
consider both speci…cations. The later …nd a model with bf > 0 …ts the investment data better than
bF > 0. That is why this paper focuses on the speci…cation of bf > 0. For given model parameters
speci…ed in Section 3, if bf = 0:05, similar results for investment policies, short run e¤ects and long run
e¤ects are found at around bF = 0:005.
7
2.3
Denote
Dynamic Optimization
(Xt ; Kt ; It ) as the net revenue of the …rm in each period t. That is
(Xt ; Kt ; It ) = (Xt ; Kt ; It )
G(Xt ; Kt ; It )
It
(9)
Assumption 8 The …rm is risk-neutral and discounts future net revenue at a constant
rate r, where r > exp ( ) 1.
As explained in Appendix A, the condition r > exp ( ) 1 guarantees a …nite …rm
value hence is one of those necessary conditions for the existence of a solution to …rm’s
optimization problem.
In each period investment is chosen to maximize the discounted present value of current and expected future net revenues, where expectations are taken over the distribution
of future demand conditions.
V (Xt ; Kt ) = max Et
It
P1
s=0
1
(Xt+s ; Kt+s ; It+s )
(1 + r)s
According to the Principle of Optimality (Theorem 9.2, Stokey and Lucas, 1989),
this investment decision can be represented as the solution to a dynamic optimization
problem de…ned by the stochastic Bellman equation
V (Xt ; Kt ) = max
It
(Xt ; Kt ; It ) +
1
Et [V (Xt+1 ; Kt+1 )]
1+r
(10)
together with the law of motion (1) and (4) for Kt and Xt . Here V (Xt ; Kt ) is the value
of the …rm in period t; Et [V (Xt+1 ; Kt+1 )] is the expected value of the …rm in period t + 1
conditional on information available in period t.
2.4
Investment Policy
In the special case of no capital adjustment costs, there is a closed-form solution that
describes the optimal investment policy analytically.
2.4.1
Frictionless Case
If G(Xt ; Kt ; It )
0, the Euler equation for the optimization problem (10) is
h
where
Xt
bt
K
=J
(11)
r+
(12)
1+r
The left hand side of equation (11) is the MPK, while the right hand side is known as
the Jorgensonian ucc. Hence despite the uncertainty about future demand, this intertemporal optimality condition is equivalent to the …rst order condition in a static decision
problem of the neoclassical producer theory. This is solely the result of the …rm being
able to adjust its capital stock instantaneously and costlessly.
J
8
Proposition 1 Investment Policy in the Frictionless Case:
The optimal frictionless investment rate is
It
Kt
=H
Xt
Kt
1
(13)
The optimal frictionless productive capital stock is
b t = It + Kt = HXt
K
where
h
J
H=
(14)
1
(15)
Proof: By investment Euler equation.
Equations (13) and (14) imply that without any friction, the optimal investment rate
is a linear function of demand relative to inherited capital stock to meet the imbalance
between the optimal productive capital stock and the level of demand in each period,
where the slope term H re‡ects production technology, demand elasticity, factor price,
and the Jorgensonian ucc.
2.4.2
Friction Cases
In the presence of general capital adjustment costs speci…ed in equation (8), there is
in general no analytical solution to the dynamic optimization problem (10). Appendix
A explains how numerical dynamic programming methods are employed to solve such
problem.
The investment model outlined above is fully parametric. Sections 2 and 3 impose
common parameter values as those in Fig. 1 of Abel and Eberly (1999). That is, depreciation rate = 0, discount rate r = 0:05, capital share = 0:33, demand elasticity " = 10,
and demand growth rate = 0:029. Given the restriction > 21 2 , the highest level
0:2405, which is denoted as the reference
of uncertainty that could be considered is
level of uncertainty in this model. Figures 1-3 present the investment policies derived
from the numerical solutions under di¤erent forms of capital adjustment costs and at half
of the reference level of uncertainty = 0:5 .
Xt
By plotting the optimal investment rate KItt against the scaled demand (H K
1), the
t
o
frictionless investment policy is a 45 line. This line is plotted as a benchmark in each of
these …gures. Since the scaled demand is a monotonic increasing transformation of the
MPK, this 45o line highlights the proposition that in the absence of any adjustment cost,
investment rate is a continuous and strictly increasing function of the MPK.
Figure 1a illustrates the investment policy with complete irreversibility only (bi = 1:0,
bq = bf = 0) that has been studied in Abel and Eberly (1999), and Figure 1b with partial
irreversibility only (bi = 0:10, bq = bf = 0) . In both these two …gures, there is a
region of inaction in the investment policy. Positive investment is triggered only when
the MPK reaches a right critical level; and at further higher levels of the MPK the
investment rate continues to be lower than what would be chosen in the frictionless case.
Since the investment rate on the 45o line would equalize the Jorgensonian ucc and the
9
MPK, this implies the introduction of irreversibility increases the ucc relative to the
Jorgensonian ucc, as highlighted in Abel and Eberly (1999). Under partial irreversibility,
no disinvestment occurs unless the MPK falls below a left critical level; and for further
lower levels of the MPK the disinvestment rate is much smaller than what would be
chosen in the frictionless case. Under complete irreversibility, no disinvestment would
ever happen, no matter how low the MPK is. To summarize, the optimal investment
policy under irreversibility is a ‘barrier control’policy and a non-decreasing function of
the MPK.
Figure 2 illustrates both a region of inaction and discontinuities in the investment
policy with …xed adjustment costs only (bf = 0:05, bi = bq = 0). Similar to partial
irreversibility, investment and disinvestment occur only when the MPK exceeds the right
and left critical levels that determine a region of inaction. Outside this region, the
optimal investment decisions are quite di¤erent from those under partial irreversibility.
Small adjustments to the capital stock do not generate bene…ts that are su¢ ciently high
to warrant paying a …xed cost to implement them. Therefore capital stock adjusts to
new information about demand through infrequent but large adjustments. When the
MPK exceeds the critical levels, optimal investment rate jumps discontinuously to an
investment policy, in which the absolute magnitude is close to or even larger than that
in the frictionless case, as the result of two countervailing e¤ects. On the one hand,
similar to irreversibility, the introduction of …xed costs increases the ucc relative to the
Jorgensonian ucc. Hence the investment rate that equalizes the ucc and the MPK would
be lower than the 45o line. On the other hand, as illustrated in Cooper, Haltiwanger and
Power (1999), with deterministic positive demand growth in this model (and/or physical
capital depreciation more generally) the …xed costs of adjustment provides an incentive
for the …rm to ‘overshoot its target’, that is whenever investment is implemented, it is
optimal to overinvest to make the MPK lower than the ucc. To summarize, the optimal
investment policy under …xed adjustment costs is a ‘jump control’ policy and a nondecreasing function of the MPK.
Figure 3 illustrates the optimal investment policy with quadratic adjustment costs
only (bq = 0:50, bi = bf = 0). Similar to the frictionless case, with quadratic adjustment
costs, investment or disinvestment takes place at all levels of the MPK. However, di¤erent
from the frictionless case, the rate of adjustment is much smaller than what would be
chosen in the frictionless case. This is because the increasing marginal adjustment costs
penalize high rates of investment and disinvestment. Capital stock thus adjusts to new
information about demand through a series of continuous but small adjustments. To
summarize, the optimal investment policy under quadratic adjustment costs is also a
continuous and strictly increasing function of the MPK, but much dampened compared
with that in the frictionless case.
3
The E¤ects of Adjustment Costs and Uncertainty
This section examines the e¤ects of uncertainty on the capital stock adjustment and the
expected capital stock level under di¤erent forms of adjustment costs. To isolate the
10
e¤ect of uncertainty, the analyses focus on changes in the distribution of demand shocks
that preserve the mean level of demand E [Xt ].
Lemma 2 Mean-preserving Spread:
Keeping constant and increasing is a mean-preserving spread for Xt , i.e. conditioning on x0 = 0,
E [Xt ] = exp ( t)
V ar [Xt ] = [exp (2 t)] exp
2
t
1
Proof: By Assumption 4.
3.1
Investment Policies at Di¤erent Level of Uncertainty
To study the e¤ects of uncertainty, it is useful to illustrate how investment policies under
di¤erent forms of adjustment costs would vary with the level of uncertainty. In addition
to the investment policies plotted at = 0:5 as those in Figures 1-3, Figures 4-6 add the
investment policies at = on the same …gures, keeping all other parameters constant.
The comparison between the dark and light lines in Figures 4-6 therefore show the e¤ects
of uncertainty on the investment policy under each form of adjustment costs.
Figure 4a and 4b consider these e¤ects under complete irreversibility and partial irreversibility. In both cases, higher level of uncertainty has two e¤ects: …rst, to enlarge
the region of inaction; and second, to lower the rate of positive investment, if positive investment would take place under both levels of uncertainty. This implies the ucc
in the presence of irreversibility is an increasing function of uncertainty, a proposition
demonstrated in Abel and Eberly (1999). Similar e¤ects are found in Figure 5 for …xed
adjustment costs as well. However, these e¤ects are di¤erent in Figure 6 for quadratic
adjustment costs, where the shape of investment policy does not vary with the level of
uncertainty, but a lower level of uncertainty implies a higher rate of investment for any
given level of MPK.
3.2
Short-run Capital Stock Adjustment
Following Bloom (2009), this section illustrates the e¤ects of adjustment costs and uncertainty on short-run investment dynamics by considering the impact e¤ect of demand
shocks et on the adjustment of the capital stock in the same period. One measure for
how much capital stock is adjusted in period t is the change in the log of capital stock
level in this period. Denote this measure as (t). That is
(t)
b t = ln K
bt
ln K
bt
ln K
1
(16)
Together with the capital accumulation formula (1), this quantity
is approximately
equal
h
i
to investment rate net of depreciation rate, i.e. (t) = ln 1 + KItt (1
) ' KItt
.
A weaker impact e¤ect indicates a smaller response of capital stock to new information
about demand, hence slower investment dynamics.
11
Lemma 3 Capital Stock Adjustment in the Frictionless Case:
If G(Xt ; Kt ; It )
0, the capital stock adjusts to demand shocks instantaneously and
fully according to a one-to-one linear relationship
(t) = e + et
(17)
Proof: By Proposition 1, Assumption 4 and equation (16).
Figures 7-9 illustrate how the level of uncertainty a¤ects this impact e¤ect under
and = 0:5 . By plotting the capital
di¤erent forms of adjustment costs, at =
stock adjustment (t) against the demand shocks et , the relationship in the frictionless
case is a 45o line. This line is plotted as a benchmark in each of these …gures. Since
1 2
, keeping constant and varying implies that e would vary with the level
e=
2
of uncertainty, which is re‡ected in the di¤erence between the dash and solid straight
lines.6 However, uncertainty only makes the di¤erence in the intercept but not in the
shape or slope of how capital stock responses to demand shocks. Therefore in the absence
of adjustment costs, the impact e¤ect of demand shocks on capital stock adjustment is
insensitive to the level of uncertainty.
With adjustment costs, Appendix C explains how other curves in Figures 7-9 are
simulated using numerical methods, so that comparison between the circle/asterisk lines
and the dash/solid 45o lines illustrates the e¤ect of adjustment costs; and comparison
between the circle line and asterisk line illustrates the e¤ect of uncertainty.
Figure 7a and 7b consider these e¤ects under complete irreversibility and partial
irreversibility. As expected from the investment policies shown in Figure 1a and 1b, the
impact e¤ect of positive demand shocks on capital stock growth is much weaker under
irreversibility than in the frictionless case. Whereas a …rm adjust instantaneously and
fully to new information about demand in the frictionless case, if the demand shock leaves
a …rm within its region of inaction, capital stock does not adjust at all in the current
period under irreversibility. If a …rm does some adjustment in the current period, the
magnitude of the adjustment is much smaller than that in the frictionless case. Also as
expected, the impact e¤ect of negative demand shocks on capital stock adjustment is
much weaker under partial irreversibility, re‡ecting the greater reluctance of the …rm to
undertake disinvestment. This impact e¤ect is exactly zero under complete irreversibility,
re‡ecting the no disinvestment constraint.
Consistent with the investment policies at di¤erent levels of uncertainty illustrated in
Figure 4a and 4b, the asterisk lines shown in Figure 7a and 7b illustrate that the impact
e¤ect of positive demand shocks on capital stock growth is noticeably stronger when the
…rm subject to irreversibility operates in a less uncertain environment, although how
capital stock responses to negative demand shocks is less distinguishable.
Figure 8 illustrates these e¤ects under …xed adjustment costs. Similar to the e¤ect
of irreversibility, if the demand shock leaves a …rm within its region of inaction, capital
stock does not adjust at all in the current period under …xed adjustment costs. Di¤erent
from the e¤ect of irreversibility, if a …rm does some adjustment in the current period,
6
This is a natural result of the unit root process de…ned in equation (4). In order to keep the mean
b t varies with .
of Xt equal to , the mean of ln Xt and hence of ln K
12
the magnitude of the adjustment is close to or even larger than that in the frictionless
case. Similar to that under irreversibility, the impact e¤ect of positive demand shocks
on capital stock growth is also much stronger when the …rm subject to …xed adjustment
costs operates in a less uncertain environment.
Figure 9 shows these e¤ects under quadratic adjustment costs. As expected from
the investment policy shown in Figure 3, the impact e¤ect of both positive and negative
demand shocks on capital stock adjustment is much weaker under quadratic adjustment
costs than in the frictionless case. Furthermore, consistent with the investment policies
at di¤erent level of uncertainty illustrated in Figure 6, the impact e¤ect is insensitive
to the level of uncertainty over the whole range of demand shocks. Similar to that in
the frictionless case, uncertainty only makes the di¤erence in the intercept but not in
the shape or slope of how capital stock responses to demand shocks under quadratic
adjustment costs.
Properties illustrated in Figures 7-9 are summarized in Proposition 2.
Proposition 2 The Short-run E¤ ect of Adjustment Costs:
If bi > 0 or bq > 0, @ (t) =@et < @ (t) =@et , 8et ;
if bf > 0, the e¤ect of @ (t) =@et relative to @ (t) =@et is ambiguous.
The Short-run E¤ ect of Uncertainty:
If bi > 0 or bf > 0, @ 2 (t) =@et @
0, 8et > 0;
2
if bq > 0, @ (t) =@et @ = 0, 8et .
3.3
Long-run Capital Stock Accumulation
Following Abel and Eberly (1999), this section illustrates the e¤ects of adjustment costs
and uncertainty
on capital stock accumulation by considering the expected capital stock
h i
b
level E Kt at di¤erent levels of uncertainty .
Lemma 4 Expected Capital Stock Level in the Frictionless Case:
If G(Xt ; Kt ; It ) 0, the expected capital stock level is given by
h i
b t = H exp ( t)
E K
(18)
Proof: By Proposition 1 and Lemma 2.
Following the Eq. (14a) in Abel and Eberly (1999), de…ne (t) as the ratio of the
expected capital stock level at date t under di¤erent forms of adjustment costs to the
expected capital stock level at date t in the frictionless case. That is
h i
bt
E K
h i
(19)
(t)
bt
E K
Lemma 4 implies the denominator in (t) is invariant to the level of uncertainty and
is a constant for given parameter values and date t. Therefore how (t) is di¤erent from
1 re‡ects the e¤ect of adjustment costs and how (t) varies with re‡ects the e¤ect of
uncertainty.
13
Figure 10 is an analytical replicate for the Fig. 1. in Abel and Eberly (1999) and
is plotted according to their analytical solution derived in the particular case: complete
irreversibility only, no depreciation, in…nite time horizon and continuous time.
Figures 11-13 illustrate how (t) varies with under di¤erent forms of adjustment
0:0485 to approximate
costs, over a range from a low level of uncertainty
=
complete certainty to the reference level of uncertainty = . Appendix C explains how
these …gures are simulated using numerical methods.
Figure 11a plots the ratio (t) against with complete irreversibility. Therefore it is a
numerical replicate for the Fig 1. in Abel and Eberly (1999) or for Figure 10. The dashed
line shows the actual estimates of (t) at di¤erent levels of , which ‡uctuate somewhat
as the result of numerical discretization. The solid line …ts a simple 3-order polynomial
regression through these points to illustrate the general pattern. This reproduces the
main features of Figure 10, which con…rms the analytical results in Abel and Eberly
(1999) and suggests that our numerical results are in the right ballpark.
There are two key features of (t) in this special case, which highlight the two important …ndings from Abel and Eberly (1999). First, (t) may be greater than, less than , or
equal to 1. Second, the behavior of (t) is not monotonic in the level of uncertainty. To
be more speci…c, at very low levels of uncertainty, the presence of complete irreversibility has almost no e¤ect on the expected level of the capital stock. Indeed as = ,
complete irreversibility becomes irrelevant
i a …rm that is experiencing certain,
h ipositive
h for
b . Over
b t initially increases relative to E K
growth in demand. As increases, E K
t
this range the ‘hangover’e¤ect described in Abel and Eberly (1999) dominates the ‘user
cost’e¤ect, so that on average
h i (t) > 1 and @ (t)=@ > 0. This
h e¤ect
i peaks at values
b t is about 1.3% higher than E K
b t . After this peak,
of around 0:17, where E K
@ (t)=@ < 0. For higher values ofh , ithe ‘user cost’ e¤ect dominates
i ‘hangover’
h the
b
b
e¤ect so that (t) < 1. At = , E Kt is about 0.3% lower than E Kt .
Figure 11b considers partial irreversibility. The relationship between (t) and has
a similar pattern to that shown under complete irreversibility, but the magnitudesh arei
bt
di¤erent. At low levels of uncertainty, (t) is again increasing in . At the peak E K
h i
b , and this peak occurs at lower values of around 0.13.
is about 0.5% higher than E K
t
At higher levels of uncertainty,
h i (t) is again decreasing
h i in . At = , the e¤ect of
b t by about 7% of E K
b t . The ‘hangover’e¤ect appears to
uncertainty is to reduce E K
be less important under partial irreversibility, where the …rm can choose to adjust capital
stock downwards, which is ruled out under complete irreversibility.
Figure 12 presents the relationship with …xed adjustment costs. First, di¤erent from
that under irreversibility, (t) is less than 1 at complete certainty in the presence of
…xed adjustment costs. A …rm with deterministic positive demand growth in this model
(and/or with physical capital depreciation more generally) will want to have growing
capital stock, which requires positive investment on average. Under …xed adjustment
costs, this adjustment will take the form of infrequent, large investments, implying a ucc
associated with …xed adjustment
than the Jorgensonian ucc. This
h i costs that his higher
i
b
b
user cost e¤ect reduces E Kt relative to E Kt by 4% at = .
14
Second, under …xed adjustment costs, as illustrated in Figure 5, a higher level of
uncertainty will …rst, enlarge the region of investment inaction. This will reduce both
investment and disinvestment relative to that under a lower level of uncertainty, hence
has an ambiguous e¤ect on the expected capital stock level; Second, outside the region
of inaction, a higher level of uncertainty will decrease the magnitude of investment and
increase the magnitude of disinvestment relative to that under a lower level of uncertainty.
This will unambiguously reduce the expected capital stock level. Finally, as illustrated
in Figure 8, a higher level of uncertainty will enlarge the support of demand shocks,
so that some larger capital adjustment which would not occur under a lower level of
uncertainty will take place under a higher level of uncertainty. However, since this implies
larger adjustment both upwards and downwards, it also has an ambiguous e¤ect on the
expected capital stock level. Taking into account all these e¤ects, how (t) varies with
under …xed adjustment costs is ambiguous. Whether (t) is larger or smaller than
1 when > 0 is therefore also ambiguous. For the case under illustration, there is an
inverse U-shapeh relationship
between h(t) iand . At = , the e¤ect of uncertainty
i
b t by about 6% of E K
b . This is similar to the magnitude found in
is to reduce E K
t
the speci…cation with partial irreversibility, and considerably larger than the e¤ect under
complete irreversibility.
Figure 13 studies a case with quadratic adjustment costs. First, similar to that under
…xed adjustment costs, the presence of quadratic adjustment costs makes
h i (t) < 1 even
b t is about 5%
in an environment with complete certainty. For example, at = , E K
h i
b . This is because with complete certainty a …rm with deterministic
lower than E K
t
positive demand growth in this model (and/or with capital depreciation) will require
positive investment. With the functional form of the quadratic adjustment costs considered here, positive investment implies that some adjustment costs must be paid, hence
a ucc associated with quadratic adjustment costs that
h iis higher than hthe iJorgensonian
b t relative to E K
b at complete
ucc. This user cost e¤ect unambiguously reduces E K
t
certainty.7
Furthermore, di¤erent from …xed adjustment costs, under quadratic adjustment costs,
(t) falls monotonically with . The magnitude of this e¤ect is also much greater than
what
or …xed adjustment costs. For = 0:15,
h has
i been found with partial irreversibility
h i
b t is about 10% lower than E K
b . At = , the e¤ect of uncertainty is to reduce
E K
t
h i
h i
b t by about 35% of E K
b . The intuition for this negative monotonic e¤ect lies in
E K
t
three facts.8 First, as illustrated in Figure 9, a higher level of uncertainty will enlarge the
1
h
bt =
Formally, if bq > 0, the closed-form Euler equation of investment implies that K
Xt ,
J+Ct
h
i
2
bq
It+1
It+1
It
1
where Ct = bq K
bq 11+r Et K
. When = 0 and > 0 (and/or
2
1+r Et
Kt+1
t
t+1
7
> 0), there must be an optimal deterministic investment rate 0 < i < 1. This simpli…es Ct = C '
1
bt < K
bt .
b t = h Xt , therefore K
J bq i > 0. Recall K
J
8
Formally, if bq > 0, when
> 0 and
> 0 (and/or > 0), the closed-form Euler equation of
1
h i
It
h
bt = E
Xt , where Ct ' J bq K
.
investment implies that E K
J+Ct
t
15
support of demand shocks, which implies larger upwards and downwards adjustment will
take place relative to that under a lower level of uncertainty. Second, di¤erent from …xed
adjustment costs, under which the adjustment cost incurred is independent of the rate of
investment, the cost incurred under quadratic adjustment costs increases monotonically
with the rate of investment. Therefore the user cost e¤ect associated with quadratic
adjustment costs increases monotonically with the level of uncertainty. Finally, there
is decreasing marginal return tohcapital
(0 < < 1). This leads to the unambiguous
i
b t and , hence between (t) and .
negative relationship between E K
3.4
The Cost-to-Pro…t Ratio
Compared with bi = 0:1, which can be interpreted as capital being sold at a price 10%
lower than the purchase price if disinvestment occurs, it is less clear how costly capital
adjustment is due to a …xed adjustment cost at the magnitude of bf = 0:05 and a quadratic
adjustment cost at the magnitude of bq = 0:50. The actual adjustment costs incurred as
a ratio of the operating pro…t provides an indication of the relative magnitude of these
costs.
to , at bf = 0:05, this cost-to-pro…t ratio increases
When
increases from
monotonically from 0.31% to 0.65%, with an average at 0.38%; at bq = 0:50, the costto-pro…t ratio increases monotonically from 0.36% to 1.56%, with an average at 0.79%.
This implies the actual adjustment costs incurred in the presence of …xed and quadratic
adjustment costs both increase with the level of uncertainty.
A related question is why the e¤ect of uncertainty on (t) appears to be much larger
in the presence of quadratic adjustment costs than that of …xed adjustment costs, at
least in the cases illustrated in Figure 13 and 12. Is it simply because that bq = 0:50
implies a higher average cost-to-pro…t ratio than that implied by bf = 0:05, or is it
because the e¤ects of uncertainty in the presence of these two forms adjustment costs are
fundamentally di¤erent, even if they would incur the same cost-to-pro…t on average?
In order to control for the …rst possibility, exercises are done to gradually increase the
value of bf and decrease the value of bq . At bf = 0:10, the cost-to-pro…t ratio increases
monotonically from 0.48% to 0.90%, with an average at 0.6%; at bq = 0:30, the cost-topro…t ratio increases monotonically from 0.23% to 1.25%, with an average at 0.6%, too.
At the median/mean level of uncertainty = 0:1445, the cost-to-pro…t ratio is about
0.55% for both bf = 0:10 and bq = 0:30.
Figure 14 plots how (t) varies with at bf = 0:10 and bq = 0:30 on the same scale.
The line for (t) associated with bf = 0:10 starts with 0.94 at = and decreases to
0.91 at = ; while the line for (t) associated with bq = 0:30 decreases from 0.97 to
0.79. And it is also around the median/mean level of uncertainty that these two lines
intersect. Finally, if the lines for (t) at bf = 0:05 and bq = 0:50 are added on the same
…gure, the line for bf = 0:10 is below the one for bf = 0:05 at any level of uncertainty;
and the line for bq = 0:30 is above the one for bq = 0:50 at any level of uncertainty.
This exercise implies …rst: in the presence of both …xed and quadratic adjustment
costs, the cost-to-pro…t
is an informative
h ratio
i
h i indicator for how much the adjustment
b
b . Second, (t) responses to in a stronger
costs would reduce E Kt relative to E K
t
16
pattern in the presence of quadratic adjustment costs than that of …xed adjustment costs,
even if the costs incurred are similar on average over the range of uncertainty. Third,
(t) is a decreasing function of bf and bq . In other words, all else being equal, higher
…xed and quadratic adjustment costs imply lower expected capital stock level.
Suppose > 0 (and/or > 0), properties discussed in this section are summarized in
Proposition 3.
Proposition 3 The Long-run E¤ ect of Adjustment Costs:
=1
if bi > 0
If = 0, (t)
;
< 1 if bf > 0 or bq > 0
<1
if bq > 0
if > 0, (t)
.
? 1 if bi > 0 or bf > 0
Furthermore, @ (t)=@bi > 0, @ (t)=@bf < 0, @ (t)=@bq < 0.
The Long-run E¤ ect of Uncertainty:
If bq > 0, @ (t)=@ < 0;
if bi > 0 or bf > 0, the sign of @ (t)=@ is ambiguous.
4
The E¤ects of Other Model Parameters
These results are obtained by using particular parameter values imposed in Abel and
Eberly (1999). This section studies for given level of uncertainty and capital adjustment
costs considered in this model, whether and how these …ndings would vary with the value
of other model parameters.
4.1
Firm’s Characteristics and Economic Environment
Following the section 5 of Abel and Eberly (1999), parameters of interest here are demand
growth rate , the discount rate r, the capital share in the production function and
the price elasticity of demand ".
One could study how the investment policy, short-run capital adjustment and longrun capital accumulation vary with each of these parameters, under each form of capital
adjustment costs. For most of these parameters, the variation is found to be most informative in the long-run capital accumulation. Therefore Figures 15-17 focus on the
long-run e¤ects only. The lines labelled as ‘AE parameters’in these …gures are plotted at
those parameter values imposed in Abel and Eberly (1999) and employed in Figures 11-13,
namely = 0:029, r = 0:05, = 0:33, and " = 10. Using these values as benchmark and
varying each of them individually in plotting other four lines provides comparative statics
in these …gures. The alternative values considered are = 0:04, r = 0:10, = 0:13, and
" = 20.
Abel and Eberly (1999) …nd that in the special case of complete irreversibility , changing in these parameters leads to clear changes in capital accumulation. To be more speci…c, …rst, @ (t)=@ < 0. This is because although the irreversibility constraint become
less important in a higher growth environment, the hangover e¤ect is weakened even more
than the user cost e¤ect as increases. Second, @ (t)=@r > 0. While the user cost under
both irreversibility and frictionless case rises with r, which tends to reduce the capital
17
stock level in both cases, this e¤ect is weaker under irreversibility than in the frictionless
1
case. Finally, @ (t)=@ < 0 and @ (t)=@" < 0. Since = 1+ ("
, as derived in equation
1)
(6), together, the capital share and the demand elasticity determine the concavity of the
pro…t function, measured by . As rises, the pro…t function becomes more concave and
thus deviations from the optimal frictionless capital stock are more costly to the …rm, so
that @ (t)=@ > 0, or equivalently @ (t)=@ < 0 and @ (t)=@" < 0. Figure 15a presents
how (t) varies with these four parameters in the presence of complete irreversibility,
which con…rms above predictions.
Di¤erent from complete irreversibility, in the presence of other forms of adjustment
costs, since there are no analytical results to draw on, the results illustrated in Figures
15b-17 are based on simulation.
4.2
Depreciation Rate
Concerning the e¤ect of depreciation rate, lines for = 0:05 are added in Figures 15-17
to compare with the benchmark case = 0. An increase in makes the investment
policy under partial irreversibility more similar to that under complete irreversibility.
For example, keeping all other parameters constant, with 5% depreciation rate, even the
partial irreversibility is at the magnitude of bi = 0:1, no disinvestment would ever happen
as if bi = 1. This is because with a higher depreciation rate, it is optimal for the …rm to sell
capital much less often. This ‘less necessary to disinvest’enhances the ‘hangover’e¤ect
under partial irreversibility, but does not a¤ect the ‘hangover’ e¤ect under completely
irreversibility. Meanwhile, a higher unambiguously increases the Jorgensonian ucc in
both cases. Therefore @ (t)=@ < 0 if bi = 1 and @ (t)=@ > 0 if bi = 0:1, as illustrated
in Figure 15a and 15b.
In the presence of …xed adjustment costs, with a higher , it is optimal for the …rm to
adjust capital stock more often than otherwise, therefore paying more adjustment costs
on average. This user cost e¤ect implies @ (t)=@ < 0 if bf > 0.
In the presence of quadratic adjustment costs, instead of simply a¤ecting the level of
(t), a higher will also dampen the e¤ect of uncertainty on (t). This is because the user
cost associated with quadratic adjustment increases with the rate of investment, while
there are two determinants for the optimal investment rate in the presence of quadratic
adjustment costs: one is the non-stochastic target level, which is an increasing function
of as demonstrated in Proposition 1; another is the stochastic optimal response to
the actual realization of demand shocks. An increase in increases the user cost by
increasing the target investment rate. Meanwhile a higher also reduces the relative
weight of the stochastic part in determining the optimal investment rate and thus the
user cost, therefore reduce the sensitivity of (t) to the level of uncertainty.
4.3
Trend-Stationary Stochastic Process
Finally, in order to allow for the demand shocks to have a persistent but not permanent
e¤ect on investment behavior, this section considers an alternative speci…cation for the
stochastic process.
18
Assumption 9 Alternative Demand Stochastic:
The law of motion for Xt is
xt = log Xt
xt = c + t +
t
where 0 <
< 1, et =
t,
=
i:i:d:
t
t 1
t
+ et =
N (0; 1) and
t
+
0
0
Xt
(20)
1
s=0
s
et
s
= 0.
Under this trend-stationary speci…cation, by adjusting the constant term c properly,
keeping constant and increasing will be a mean-preserving spread for Xt as well.
Lemma 5 Alternative Mean-preserving Spread:
2
For any 0 < < 1, and t ! 1, by choosing c = 0:5 2 = (1
), keeping constant
and increasing is a mean-preserving spread for Xt , i.e. conditioning on 0 = 0,
E [Xt ] = exp ( t)
V ar [Xt ] = [exp (2 t)] exp
2
= 1
2
1
Proof: By Assumption 9.
By construction, the stationary process (20) therefore implies the same expected value
for Xt as predicted in Lemma 2 for the non-stationary process (4).9 Appendix B explains
how the dynamic optimization problem (10) can be solved numerically under this alternative speci…cation. When 0 < < 1, similar patterns for the investment policy, short-run
capital adjustment and long-run capital accumulation are found as those illustrated in
previous sections. Finally, the relationships between (t) and derived at = 0:9 are
added in Figures 15-17 to illustrate the e¤ects of adjustment costs and uncertainty on
capital accumulation under a trend-stationary stochastic process.
5
Conclusions
Investment is one of the most important and volatile components of macroeconomic
activity. In the short run, the relationship between uncertainty and investment is central
to understanding the business cycle. In the long run, the e¤ect of uncertainty on capital
accumulation has signi…cant implications for economic growth and development. This
paper o¤ers a uni…ed framework to study how capital adjustment costs and uncertainty
a¤ect investment dynamics and capital accumulation.
From complete and partial irreversibility, to …xed and quadratic adjustment costs, a
simple investment model could generate very rich implications for di¤erent investment
behavior according to di¤erent forms of adjustment costs. The impact e¤ects of demand
shocks on capital adjustment also dramatically di¤er depending on the form of adjustment
9
Although (4) can be regarded as the limit case of (20) when
! 1, the e¤ects of uncertainty
generated from (4) and (20) are not completely comparable in general. This is because for a given
E [Xt ], V ar [Xt ] is time-invariant under Assumption 9, but will increase with t under Assumption 4.
19
costs; however, these e¤ects may be ampli…ed, dampened or even reversed when translated
into the expected long-run capital stock level. Other parameters in the model would also
a¤ect the quantities of interest in a substantial fashion.
These …ndings therefore highlight the importance of empirical work. To obtain the
right sign and quantify the magnitude for the e¤ects of uncertainty, it is therefore important to study which form of capital adjustment costs could explain the investment
behavior best, conditioning on a good control for the …rm’s characteristics and economic environment. This is consistent with the direction in recent empirical work relying
on structural estimation and micro-level data, such as Cooper and Haltiwanger (2006),
Eberly, Rebelo and Vincent (2008), Bond, Söderbom and Wu (2008) and Bloom (2009).
20